Difference between revisions of "Smooth map"
From Maths
(Created page with "==Definition== A map {{M|f:M\rightarrow N}} between two smooth manifolds {{M|(M,\mathcal{A})}} and {{M|(N,\mathcal{B})}} (of not necessarily the same dimen...") |
m |
||
| Line 1: | Line 1: | ||
| + | '''Note: ''' not to be confused with [[Smooth function|smooth function]] | ||
| + | |||
==Definition== | ==Definition== | ||
A map {{M|f:M\rightarrow N}} between two [[Smooth manifold|smooth manifolds]] {{M|(M,\mathcal{A})}} and {{M|(N,\mathcal{B})}} (of not necessarily the same dimension) is said to be smooth<ref>Introduction to smooth manifolds - John M Lee - Second Edition</ref> if: | A map {{M|f:M\rightarrow N}} between two [[Smooth manifold|smooth manifolds]] {{M|(M,\mathcal{A})}} and {{M|(N,\mathcal{B})}} (of not necessarily the same dimension) is said to be smooth<ref>Introduction to smooth manifolds - John M Lee - Second Edition</ref> if: | ||
| − | * <math>\forall p\in M\exists\ (U,\varphi)\in\mathcal{A},\ p\in U\text{ and }(V,\psi)\in\mathcal{B}</math> such that <math>F(U)\subseteq V\wedge \psi\circ F\circ\varphi^{-1}</math> is [[Smooth|smooth]] | + | * <math>\forall p\in M\exists\ (U,\varphi)\in\mathcal{A},\ p\in U\text{ and }(V,\psi)\in\mathcal{B}</math> such that <math>F(U)\subseteq V\wedge[\psi\circ F\circ\varphi^{-1}:\varphi(U)\rightarrow\psi(V)]</math> is [[Smooth|smooth]] |
| + | |||
| + | ==See also== | ||
| + | * [[Smooth function]] | ||
| + | * [[Smooth manifold]] | ||
| + | |||
==References== | ==References== | ||
<references/> | <references/> | ||
{{Definition|Manifolds}} | {{Definition|Manifolds}} | ||
Revision as of 20:44, 13 April 2015
Note: not to be confused with smooth function
Definition
A map [ilmath]f:M\rightarrow N[/ilmath] between two smooth manifolds [ilmath](M,\mathcal{A})[/ilmath] and [ilmath](N,\mathcal{B})[/ilmath] (of not necessarily the same dimension) is said to be smooth[1] if:
- [math]\forall p\in M\exists\ (U,\varphi)\in\mathcal{A},\ p\in U\text{ and }(V,\psi)\in\mathcal{B}[/math] such that [math]F(U)\subseteq V\wedge[\psi\circ F\circ\varphi^{-1}:\varphi(U)\rightarrow\psi(V)][/math] is smooth
See also
References
- ↑ Introduction to smooth manifolds - John M Lee - Second Edition
